On the null-distribution of theF-statistics for testing a ‘partial’ null-hypothesis in a randomized partially balanced incomplete block design withm associate classes under the Neyman model

In a previous paper [1], it has been shown that for a partially balanced incomplete block design with two associate classes the nulldistribution of the F-statistic under the ' to ta l ' null-hypothesis (i.e., treatment-effects being all equal to zero) can be approximated by the familiar central F-distribution even under the Neyman model (i.e., an intra-block analysis model with both the unit errors and the technical errors), if it is randomized. As was announced in that paper, the approximate distributions of the F-statistic under the 'par t ia l ' null-hypothesis have been left to fur ther discussion. In the present article, the authors are concerned with this problem. They set forth the problem for a partially balanced incomplete block design with m associate classes and consider the null-distribution of the F-statistic for testing a 'par t ia l ' null-hypothesis, so that it includes the ' t o t a l ' null-hypothesis as a special case and they reached the conclusion that the null-distribution of the F-statistic can be approximated, after the randomization, by a certain central F-distribution with appropriate degrees of freedom, if certain uniformity conditions are imposed on the unit errors and the number b of the blocks is sufficiently large. In section 1 the spectral decomposition of the matrix NN', where N being the incidence matrix of the design under consideration, is given and this is useful for the later discussions. The null-distribution of the F-statistic for testing a partial nullhypothesis before the randomization under the Neyman model is presented in section 2, and this turns out to be a non-central F-distribution whose non-centrality parameter depends upon the quantities ~ and ~ both being the quadratic forms of the unit errors.